## MB03PD

### Matrix rank determination by incremental condition estimation (row pivoting)

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To compute (optionally) a rank-revealing RQ factorization of a
real general M-by-N matrix  A,  which may be rank-deficient,
and estimate its effective rank using incremental condition
estimation.

The routine uses an RQ factorization with row pivoting:
P * A = R * Q,  where  R = [ R11 R12 ],
[  0  R22 ]
with R22 defined as the largest trailing submatrix whose estimated
condition number is less than 1/RCOND.  The order of R22, RANK,
is the effective rank of A.

MB03PD  does not perform any scaling of the matrix A.

```
Specification
```      SUBROUTINE MB03PD( JOBRQ, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
\$                   RANK, SVAL, DWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER          JOBRQ
INTEGER            INFO, LDA, M, N, RANK
DOUBLE PRECISION   RCOND, SVLMAX
C     .. Array Arguments ..
INTEGER            JPVT( * )
DOUBLE PRECISION   A( LDA, * ), SVAL( 3 ), TAU( * ), DWORK( * )

```
Arguments

Mode Parameters

```  JOBRQ   CHARACTER*1
= 'R':  Perform an RQ factorization with row pivoting;
= 'N':  Do not perform the RQ factorization (but assume
that it has been done outside).

```
Input/Output Parameters
```  M       (input) INTEGER
The number of rows of the matrix A.  M >= 0.

N       (input) INTEGER
The number of columns of the matrix A.  N >= 0.

A       (input/output) DOUBLE PRECISION array, dimension
( LDA, N )
On entry with JOBRQ = 'R', the leading M-by-N part of this
array must contain the given matrix A.
On exit with JOBRQ = 'R',
if M <= N, the upper triangle of the subarray
A(1:M,N-M+1:N) contains the M-by-M upper triangular
matrix R;
if M >= N, the elements on and above the (M-N)-th
subdiagonal contain the M-by-N upper trapezoidal matrix R;
the remaining elements, with the array TAU, represent the
orthogonal matrix Q as a product of min(M,N) elementary
reflectors (see METHOD).
On entry and on exit with JOBRQ = 'N',
if M <= N, the upper triangle of the subarray
A(1:M,N-M+1:N) must contain the M-by-M upper triangular
matrix R;
if M >= N, the elements on and above the (M-N)-th
subdiagonal must contain the M-by-N upper trapezoidal
matrix R;
the remaining elements are not referenced.

LDA     INTEGER
The leading dimension of the array A.  LDA >= max(1,M).

JPVT    (input/output) INTEGER array, dimension ( M )
On entry with JOBRQ = 'R', if JPVT(i) <> 0, the i-th row
of A is a final row, otherwise it is a free row. Before
the RQ factorization of A, all final rows are permuted
to the trailing positions; only the remaining free rows
are moved as a result of row pivoting during the
factorization.  For rank determination it is preferable
that all rows be free.
On exit with JOBRQ = 'R', if JPVT(i) = k, then the i-th
row of P*A was the k-th row of A.
Array JPVT is not referenced when JOBRQ = 'N'.

RCOND   (input) DOUBLE PRECISION
RCOND is used to determine the effective rank of A, which
is defined as the order of the largest trailing triangular
submatrix R22 in the RQ factorization with pivoting of A,
whose estimated condition number is less than 1/RCOND.
RCOND >= 0.
NOTE that when SVLMAX > 0, the estimated rank could be
less than that defined above (see SVLMAX).

SVLMAX  (input) DOUBLE PRECISION
If A is a submatrix of another matrix B, and the rank
decision should be related to that matrix, then SVLMAX
should be an estimate of the largest singular value of B
(for instance, the Frobenius norm of B).  If this is not
the case, the input value SVLMAX = 0 should work.
SVLMAX >= 0.

TAU     (output) DOUBLE PRECISION array, dimension ( MIN( M, N ) )
On exit with JOBRQ = 'R', the leading min(M,N) elements of
TAU contain the scalar factors of the elementary
reflectors.
Array TAU is not referenced when JOBRQ = 'N'.

RANK    (output) INTEGER
The effective (estimated) rank of A, i.e. the order of
the submatrix R22.

SVAL    (output) DOUBLE PRECISION array, dimension ( 3 )
The estimates of some of the singular values of the
triangular factor R:
SVAL(1): largest singular value of
R(M-RANK+1:M,N-RANK+1:N);
SVAL(2): smallest singular value of
R(M-RANK+1:M,N-RANK+1:N);
SVAL(3): smallest singular value of R(M-RANK:M,N-RANK:N),
if RANK < MIN( M, N ), or of
R(M-RANK+1:M,N-RANK+1:N), otherwise.
If the triangular factorization is a rank-revealing one
(which will be the case if the trailing rows were well-
conditioned), then SVAL(1) will also be an estimate for
the largest singular value of A, and SVAL(2) and SVAL(3)
will be estimates for the RANK-th and (RANK+1)-st singular
values of A, respectively.
By examining these values, one can confirm that the rank
is well defined with respect to the chosen value of RCOND.
The ratio SVAL(1)/SVAL(2) is an estimate of the condition
number of R(M-RANK+1:M,N-RANK+1:N).

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension ( LDWORK )
where LDWORK = max( 1, 3*M ),           if JOBRQ = 'R';
LDWORK = max( 1, 3*min( M, N ) ), if JOBRQ = 'N'.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  The routine computes or uses an RQ factorization with row
pivoting of A,  P * A = R * Q,  with  R  defined above, and then
finds the largest trailing submatrix whose estimated condition
number is less than 1/RCOND, taking the possible positive value of
SVLMAX into account.  This is performed using an adaptation of the
LAPACK incremental condition estimation scheme and a slightly
modified rank decision test.

The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(k), where k = min(m,n).

Each H(i) has the form

H = I - tau * v * v'

where tau is a real scalar, and v is a real vector with
v(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit
in A(m-k+i,1:n-k+i-1), and tau in TAU(i).

The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth row of P is the ith canonical unit vector.

```
References
```   Bischof, C.H. and P. Tang.
Generalizing Incremental Condition Estimation.
LAPACK Working Notes 32, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-132,
May 1991.

 Bischof, C.H. and P. Tang.
Robust Incremental Condition Estimation.
LAPACK Working Notes 33, Mathematics and Computer Science
Division, Argonne National Laboratory, UT, CS-91-133,
May 1991.

```
Numerical Aspects
```  The algorithm is backward stable.

```
```  None
```
Example

Program Text

```*     MB03PD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER        ( ZERO = 0.0D0, ONE = 1.0D0 )
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX, MMAX
PARAMETER        ( NMAX = 10, MMAX = 10 )
INTEGER          LDA
PARAMETER        ( LDA = NMAX )
INTEGER          LDTAU
PARAMETER        ( LDTAU = MIN(MMAX,NMAX) )
INTEGER          LDWORK
PARAMETER        ( LDWORK = 3*MMAX )
*     .. Local Scalars ..
CHARACTER*1      JOBRQ
INTEGER          I, INFO, J, M, N, RANK
DOUBLE PRECISION RCOND, SVAL(3), SVLMAX
*     ..
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), DWORK(LDWORK), TAU(LDTAU)
INTEGER          JPVT(MMAX)
*     .. External Subroutines ..
EXTERNAL         MB03PD
*     .. Intrinsic Functions ..
INTRINSIC        MIN
*     .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
*     Skip the heading in the data file and read the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) M, N, JOBRQ, RCOND, SVLMAX
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) N
ELSE
IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99971 ) M
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,M )
*           RQ with row pivoting.
DO 10 I = 1, M
JPVT(I) = 0
10       CONTINUE
CALL MB03PD( JOBRQ, M, N, A, LDA, JPVT, RCOND, SVLMAX, TAU,
\$                   RANK, SVAL, DWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99995 ) RANK
WRITE ( NOUT, FMT = 99994 ) ( JPVT(I), I = 1,M )
WRITE ( NOUT, FMT = 99993 ) ( SVAL(I), I = 1,3 )
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' MB03PD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from MB03PD = ',I2)
99995 FORMAT (' The rank is ',I5)
99994 FORMAT (' Row permutations are ',/(20(I3,2X)))
99993 FORMAT (' SVAL vector is ',/(20(1X,F10.4)))
99972 FORMAT (/' N is out of range.',/' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
END
```
Program Data
``` MB03PD EXAMPLE PROGRAM DATA
6     5     R  5.D-16     0.0
1.    2.    6.    3.    5.
-2.   -1.   -1.    0.   -2.
5.    5.    1.    5.    1.
-2.   -1.   -1.    0.   -2.
4.    8.    4.   20.    4.
-2.   -1.   -1.    0.   -2.
```
Program Results
``` MB03PD EXAMPLE PROGRAM RESULTS

The rank is     4
Row permutations are
2    4    6    3    1    5
SVAL vector is
24.5744     0.9580     0.0000
```